Supplementary Material
Tunable Spatial Heterogeneity in Structure and Composition within
Aqueous Microfluidic Droplets
Su Hui Sophia Lee1, Pengzhi Wang2, Swee Kun Yap2, T. Alan Hatton1,3 and Saif A. Khan1,2*
1
Chemical and Pharmaceutical Engineering Programme, Singapore-MIT Alliance, National
University of Singapore, Singapore 117576.
2
Department of Chemical and Biomolecular Engineering, National University of Singapore, 4
Engineering Drive 4, Singapore 117576.
3
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139.
*Corresponding author: Dr. Saif A. Khan
Fax: (65) 6779 1936; Tel: (65) 6516 5133; E-mail: chesakk@nus.edu.sg
1
FIG. S1. Scaled version of the actual microchannel pattern.
2
FIG. S2. (i) Stereomicroscopic images of reticulate structures of ATPS drops (CDEX = 3.7% w/w,
CPEG = 4.0% w/w) captured at x ~ 93.9 mm with flow speeds (a) 2.1 mm/s (b) 4.3 mm/s (c) 6.4
mm/s (d) 10.7 mm/s (e) 15.0 mm/s. Scale bars represent 100 m. (ii) corresponding FFT results.
Scale bars represent the characteristic frequency, f = 20.
3
Viscosity measurements were performed using Rheometer AR G2.
FIG. S3. Equilibrium viscosities at various compositions (a) DEX-rich phase, d. (b) PEG-rich
phase,c. (c) viscosity ratio, p ( = d/c). (d) Interfacial tension, , between PEG and DEX at
various compositions. Grey and black circles represent data approximated from Ryden et al. 1 and
Helfrich et al.2 respectively (CPEG in these data are close to values of CDEX), and white circles
represent data interpolated for our calculations.
4
Comparison with established theory:
Khakhar and Ottino outline the theory for breakup of an infinitely long liquid cylinder in
unconfined, linear Stokes flows.3 We compare the characteristic size of the fluid filament from FFT
with the critical thread diameter (Dcrit) obtained from the theoretical treatment by Khakhar and
Ottino using an overestimation of the elongation rate (  ).3-5 The key differences between our work
and Khakhar and Ottino’s are that our flow field is non-linear and confined, while their flow system
is linear and unconfined. From previous studies, elongation flow is the most efficient for breakup of
liquid threads and drops.5-8 The analysis by Khakhar and Ottino assumes a liquid cylinder immersed
in an immiscible fluid, which is subjected to capillary instabilities at the interface. In an extending
thread, breakup occurs when the amplitude of a disturbance exceeds the mean thread radius that is
constantly decreasing with time. Rcrit refers to the critical thread radius at which the disturbance
starts to grow and eventually leads to breakup;5 it characterizes the minimum radius of the filament
that can possibly exist in a linear flow. Following Janssen et al., we assume Rcrit to be
approximately similar to the size of broken up drops (Rdrops).5 The results by Janssen et al. are
reproduced in Fig. S4,4,5 which show that the Rdrops is dependent on parameters such as viscosity of
the continuous phase (c), the elongation rate (  ), interfacial tension between the fluid phases ()
and viscosity ratio (p). c,  and p can be obtained from Fig. S3 and the elongation rate was
estimated by taking  ~U / h (the maximum elongation rate in the droplets), where U is flow speed
and h is channel height. The calculated Rcrit (from Fig. S4a and S4b) should characterize the finest
possible filament radius in our microfluidic droplet. The initial disturbance, 0, is taken to be 10-9
m.9 Assuming that the filament radii of reticulate structures, Rcrit, are approximately equivalent to
Rdrops, the full curve for p = 3.6 (p ~ 5.2 from Fig. S3 for CDEX = 3.7% w/w, CPEG = 4.0% w/w) from
Fig. S4a was used to estimate Rcrit(CDEX = 3.7% w/w, CPEG = 4.0% w/w) at various flow speeds, U. Similarly,
Rcrit(CDEX = 4.5% w/w, CPEG = 5.0% w/w) was estimated using the full curve corresponding to p = 10 (p ~ 15
from Fig. S3 for CDEX = 4.5% w/w, CPEG = 5.0% w/w) from Fig. S4b. Following this, we multiplied
5
Rcrit by a factor of 2 to obtain Dcrit, which characterize the finest possible filament sizes in our
microfluidic droplet.
FIG. S4 (a) Experimental (symbols) and calculated (curves) drop sizes as a function of the flow
parameter c /  where = 10-9 m (full curves) or 10-10 m (dashed curves).4 (b) Dimensionless
drop radius resulting from thread breakup during stretching as a function of the dimensionless
stretching rate at a constant rate
 .4
6
Calculation of interface thickness at equilibrium:
We evaluated the interface thickness between PEG-rich phase and DEX-rich phase by adapting
similar formulations proposed by Cahn and Hilliard for binary systems.10 The mathematical
expression for the total free energy of a ternary mixture11 is:
K
K
2
2

G    g 1 , 2   1  1   K 12  1  2   2  2   dV
2
2

V 
 (1)
where 1, 2 and 3 denote PEG, DEX and water, 1, 2 and 3 are the volume fractions of PEG, DEX
and water respectively, g(1, 2) represents the free energy of a uniform mixture of composition (1,
2, 1-1-2), and K represent gradient energy parameters (K1, K2, K12). The Flory-Huggins
expression for g(1, 2):
2
 1

RT  N ln 1  N ln 2  1  1  2  ln 1  1  2   1212  131 1  1  2  
g 1 ,2  
1
2

Vm 

   232 1  1  2 

where Vm is the molar volume of water, N1 and N2 represent the degree of polymerization. (i.e. N1 =
180; N2 = 3086).
7
The physical data applied for the analysis are summarized in the following table:
Properties
PEG(1)
DEX(2)
Molecular weight
8000
500,000
Monomer weight
44
162
Number of monomers
180
3086
Radius of gyration, RG
RG1 = ~ 3 nm12
RG2 = ~ 20 nm13
Table 1. Physical data for PEG and DEX.
The Flory-Huggins parameters14 are:
12 
245
RT
13 
100
RT
 23 
0
RT
From the theoretical treatment by Ariyapadi and Nauman,15 the gradient energy parameters can be
expressed as follows:
K12 
K1 
2


RT RG1
1
1

 13 

Vm 3 
N11  1  1  2  
K2 

RT RG2 2 
1
1

  23 

Vm 3 
N 22  1  1  2  
2

 RG2 2 

RT  RG1
1
1






 12 





13
12
23
Vm  6 
N1 1  1  2   6 
N 2 1  1  2   
According to the formulation of Cahn and Hilliard,10 the specific surface free energy,  for an
unconfined fluid domain is minimized when the g term and the concentration gradient term are
8
equal. It can be proved that this condition still holds true for a ternary polymer-polymer-solvent
system. The specific interfacial free energy for a planar interface is defined as
2
2

K1  d1 
 d1  d2  K 2  d2  
    g (1 , 2 )  
  K12 



  dx

2
dx
dx
dx
2
dx






 
 

 (2)
where ∆g(1, 2) = g0(1, 2) – 1µ1 – 2µ2 – (1-1-2)µ3 with µ1, µ2 and µ3 as the respective
chemical potentials of the three species in a uniform mixture of compositions ( 1α, 2α, 1- 1α -2α).
Here, α refers to the DEX-rich phase and β refers to the PEG-rich phase at equilibrium. The
equation above can be written in a compact form,


1  d  d    K1

g
(

)




:
 
2
dx
dx

  K12


K12  
  dx
K 2  
  3
where
 1 

 2 
 
At equilibrium, the Euler-Lagrange condition has to be satisfied, i.e.,
I d 
I

 dx   d  / dx







with I representing the integrand of Eq. (3). Since I does not explicitly depend on x, the EulerLagrange condition can be written as
d   d  
I
I 

dx   dx   d  / dx




0


 (4)
9
where
dI I d  I d 2 
I
 
  2
dx x dx  dx  d  / dx


With the boundary condition that the ∆g term and the concentration gradient tend to zero as x tends
to infinity, the Euler-Lagrange condition (Eq. (4)) yields an expression for the specific interfacial
free energy:
2
 K1  d1 2
 d1  d2  K 2  d2  
 2   
 K12 



  dx
 2  dx 
 dx  dx  2  dx  
 

The experimental data applied for our calculation were adopted from Ryden and Albertsson as
shown in table 2,1
Total
Concentration in Concentration in
concentration, bottom phase (), top
phase
(% w/w)
(% w/w)
(% w/w)
DEX
5.2
9.46
1.05
PEG
3.8
1.85
5.7
Water
91
88.69
93.25
(),
Table 2. Partitioning of PEG and DEX in the aqueous two phases.
Both composition profiles of the two polymers are assumed to vary in space as a hyperbolic tangent
function with a characteristic length ‘l’:
i  i 
i  i 
2
 x 
 1  tanh   
 l 

Ryden and Albertsson have determined the interfacial tension between the two phases by the
method of rotating drops for an overall composition of 5.2% DEX and 3.8% PEG to be around 2
10
N/m.1 These mass concentrations were converted to volume fractions through density of PEG
(~1000 kg/m3) and partial molar volume of DEX (~300 000 cm3/mol).16,17 The interfacial thickness,
l, is defined according to Cahn and Hilliard formulation10, and l is obtained by fitting the specific
interfacial free energy ( with experimental interfacial tension. In this case, l is found to be around
~4.5 m.
References:
1
J. Ryden and P. A. Albertsson, J. Colloid. Interf. Sci. 37, 219 (1971).
2
M. R. Helfrich, M. El-Kouedi, M. R. Etherton, and C. D. Keating, Langmuir 21, 8478
(2005).
3
D. V. Khakhar and J. M. Ottino, Int. J. Multiphase Flow 13, 71 (1987).
4
J. M. H. Janssen and H. E. H. Meijer, J. Rheol. 37, 597 (1993).
5
J. M. H. Janssen, Dynamics of Liquid-Liquid Mixing. Ph.D. Thesis. Eindhoven University
of Technology (1993).
6
D. V. Khakhar and J. M. Ottino, J. Fluid Mech. 166, 265 (1986).
7
B. J. Bentley and L. G. Leal, J. Fluid Mech. 167, 241 (1986).
8
H. P. Grace, Chem. Eng. Commun. 14, 225 (1982).
9
W. Kuhn, Kolloid Z. 132, 84 (1953).
10
J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958).
11
D. Q. He, S. Kwak, and E. B. Nauman, Macromol. Theory Simul. 5, 801 (1996).
12
S. Kawaguchi, G. Imai, J. Suzuki, A. Miyahara, T. Kitano, and K. Ito, Polymer 38, 2885
(1997).
13
C. E. Ioan, T. Aberle, and W. Burchard, Macromolecules 34, 326 (2001).
14
H. O. Johansson, G. Karlstrom, F. Tjerneld, and C. A. Haynes, J. Chromatogr. B 711, 3
(1998).
11
15
M. V. Ariyapadi and E. B. Nauman, J. Polym. Sci. Pol. Phys. 28, 2395 (1990).
16
A. Eliassi, H. Modarress, and G. A. Mansoori, J. Chem. Eng. Data 43, 719 (1998).
17
O. V. Alekseeva, O. V. Rozhkova, O. V. Eliseeva, and A. N. Prusov, Russ. J. Appl. Chem.
78, 971 (2005).
12